How do you feel about electronic viewfinders? Cameras without tend to be smaller (and less expensive, depending on which system you're looking at), but it can be a big step if you're used to SLRs with optical viewfinders. What are you using today, or is this a first "nice" camera? I would look at:
Olympus E-P5 (looks like it's right at £700 with the kit lens): Small, great image quality, Micro Four-Thirds lens system, which give you lots of great options that are reasonably priced. The Panasonic 20mm or Olympus 17mm would give you something like a "classic" street photography lens, and you can get that Panasonic pretty cheap (not sure if it has banding issues on the E-P5, though -- I'd research that if you're considering it).
Fuji X-E2 (a little bit over your price range new with kit lens): Also small, excellent image quality with that "Fuji look." More limited lens selection, particularly if you're looking at the less expensive end of the spectrum (<$500).
There are some fixed-lens options as well (X100), but unless you've got multiple cameras, a fixed-lens option is limiting, and you can put together something similar with the right lens and one of the bodies above. There are quite a few mirrorless options that would be great for street shooting and versatile enough for just about anything else, so you've got a lot of good options these days.